Numerical linear system solving with parametric entries by error correction

Abstract

We consider the problem of solving a full rank consistent linear system A(u)x = b(u) where the m × n matrix A and the m-dimensional vector b has entries that are polynomials in u over a field. We give an algorithm that computes the unique solution x = f (u)/g(u), which is a vector of rational functions, by evaluating the parameter u at distinct points. Those points ξ λ where the matrix A evaluates to a matrix A(ξ λ ), with entries over the scalar field, of lower rank, or in the numeric setting to an ill-conditioned matrix, are not identified but accounted for by error-correcting code techniques. We also correct true errors where the evaluation at some u = ξ λ results in an erroneous, possibly full rank consistent and well-conditioned scalar linear system. Our algorithm generalizes Welch/Berlekamp decoding of Reed/Solomon error correcting codes and their numeric floating point counterparts.We have implemented our algorithms with floating point arithmetic. For the determination of the exact numerator and denominator degrees and number of errors we use singular values based numeric rank computations. The arising linear systems for the error-corrected parametric solution are demonstrated to be well-conditioned even when the input scalars have noise. In several initial experiments we have shown that our approach is numerically stable even for larger systems m = n = 100, provided the degrees in the solution are small (≤ 2). For smaller systems m = n = 10 with higher degrees (≤ 20) the algorithm works similarly to rational function recovery. Our implementation can correct 13 true errors in both settings.

Polynomial Linear System Solving with Random Errors

“…As in [3,9], this paper focuses on a scenario in which the nodes could make errors, possibly computing ≠ ( ). In this case, the master node performs a vector Cauchy interpolation with errors in order to recover the solution ( ).…”

“…recovering the solution ( ) of the PLS given its evaluations, some of which are erroneous, is what we call Polynomial Linear System Solving with Errors (PLSwE). In order to solve PLSwE, in [3,9] the authors generalize the polynomial interpolation with errors (i.e. decoding Reed-Solomon (RS) codes) to rational functions interpolation with errors.…”

“…The goal is to minimize the number of evaluation points needed to recover the solution or equivalently to maximize the bound on the number of errors (decoding radius) that we could correct. In [3,9] they can correct up to the unique decoding radius, similarly to classical RS codes. Moreover, in [9] they show a second way to bound by exploiting the linear algebra setting as in [5].…”

LU Factorization with Errors

“…Fault-tolerant computer algebra Some recent progress on fault tolerant algorithms has been made for Chinese remaindering [21,28,3], system solving [5,25], matrix multiplication and inversion [30,20,32], and function recovery [8,26].…”

Error Correction in Fast Matrix Multiplication and Inverse

“…This is a strictly stronger model than verification, where we seek not only to identify when a result is incorrect, but also to compute the correct result if it is "close" to the given, incorrect one. Some recent errorcorrection problems considered in the computer algebra literature include Chinese remaindering [Goldreich, Ron, and Sudan, 1999, Khonji, Pernet, Roch, Roche, and Stalinski, 2010, Böhm, Decker, Fieker, and Pfister, 2015, system solving [Boyer andKaltofen, 2014, Kaltofen, Pernet, Storjohann, andWaddell, 2017], and function recovery [Comer, Kaltofen, andPernet, 2012, Kaltofen andYang, 2013].…”